Numerical Descriptive Measures Chapter 2 Borrowed from 321%20ppts/c3.ppt.
Descriptive Statistics: Numerical Measures Distribution
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Transcript of Descriptive Statistics: Numerical Measures Distribution
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1 Slide
Descriptive Statistics: Numerical Measures
Distribution
Chapter 3BA 201
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2 Slide
DISTRIBUTION
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3 Slide
Measures of Distribution Shape,Relative Location, and Detecting Outliers
Distribution Shape z-Scores Chebyshev’s
Theorem Empirical Rule Detecting Outliers
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4 Slide
Distribution Shape: Skewness An important measure of the shape of a
distribution is called skewness. The formula for the skewness of sample data is
3
)2)(1(Skewness
sxx
nnn i
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5 Slide
Distribution Shape: Skewness Symmetric (not skewed)
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = 0
• Skewness is zero.• Mean and median are equal.
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6 Slide
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Distribution Shape: Skewness Moderately Skewed Left
Skewness = .31
• Skewness is negative.• Mean will usually be less than the median.
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7 Slide
Distribution Shape: Skewness Moderately Skewed Right
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = .31
• Skewness is positive.• Mean will usually be more than the median.
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8 Slide
Distribution Shape: Skewness Highly Skewed Right
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = 1.25
• Skewness is positive (often above 1.0).• Mean will usually be more than the median.
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9 Slide
Distribution Shape: Skewness
Apartment Rents425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615
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10 Slide
Rela
tive
Freq
uenc
y
.05
.10
.15
.20
.25
.30
.35
0
Skewness = 0.92
Distribution Shape: Skewness
Apartment Rents
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11 Slide
z-Scores
z x xsii
The z-score is often called the standardized value.It denotes the number of standard deviations a data value xi is from the mean.
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12 Slide
z-Scores An observation’s z-score is a measure of the relative location of the observation in a data set.
x
z-score < 0
z-score = 0
z-score > 0
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13 Slide
• z-Score of Smallest Value (425)425 490.80 1.2054.74
ix xzs
z-Scores
Standardized Values for Apartment Rents-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
Apartment Rents
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14 Slide
PRACTICEZ-SCORES
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15 Slide
Practice #6 – z-Scores
z-Score3 -107 -611 -216 318 523 10
ix xxi x = 13s = 7.4
sxxz i
i
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16 Slide
Chebyshev’s TheoremAt least (1 - 1/k2) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1.
Within k standard
deviations of mean
% of data values
2 75%3 89%4 94%
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17 Slide
Chebyshev’s Theorem
Let z = 1.5 with = 490.80 and s = 54.74x
At least (1 1/(1.5)2) = 1 0.44 = 0.56 or 56%of the rent values must be betweenx - k(s) = 490.80 1.5(54.74) = 409
andx + k(s) = 490.80 + 1.5(54.74) = 573
(Actually, 86% of the rent values are between 409 and 573.)
Apartment Rents
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18 Slide
Empirical Rule
When data approximate a bell-shaped distribution, the empirical rule can be used to determine the percentage of data values that must be within a specified number of standard deviations of the mean.
Within … of the mean
% of data values
+/- 1 standard deviation 68.26%
+/- 2 standard deviations 95.44%
+/- 3 standard deviations 99.72%
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19 Slide
Empirical Rule
xm – 3s m – 1s
m – 2sm + 1s
m + 2sm + 3sm
68.26%95.44%99.72%
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20 Slide
PRACTICECHEBYSHEV’S THEOREM AND EMPIRICAL RULE
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21 Slide
Practice #7 - Chebyshev’s Theorem
x = 1200s = 110
k = 1.25
k = 3.5
How many items (%) are within k standard deviations?
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22 Slide
Practice #7 – Empirical Rule
x = 1200s = 110
What is the lower bound for 2 standard deviations? The upper bound? How many items (%) are within this area?
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23 Slide
Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be:• an incorrectly recorded data value• a data value that was incorrectly included in the
data set• a correctly recorded data value that belongs in
the data set
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24 Slide
Detecting Outliers
• The most extreme z-scores are -1.20 and 2.27• Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set.
-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27
Standardized Values for Apartment Rents
Apartment Rents
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25 Slide